Limits of canonical forms on towers of Riemann surfaces
We prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence of finite Galois covers of a hyperbolic Riemann Surface , converging to the universal cover. The theorem states that the sequence of forms on inherited from the canonical forms on ’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet type theorem in the context of arbitrary infinite Galois covers.
A compact connected Riemann surface of genus can be given a canonical (Arakelov) -form by embedding the surface inside its Jacobian via the Abel-Jacobi map, and pulling back the canonical (‘Euclidean’) translation-invariant -form. A celebrated theorem of Kazhdan ([Kaj, §3], see also [mumford, Kazhdan]) states that ‘the hyperbolic form is the limit of the canonical forms’: if is an ascending sequence of finite Galois covers converging to the universal cover, then the -forms on inherited from canonical forms via converge uniformly to a multiple of the hyperbolic -form. See [rhodes1993sequences] and [mcmullen2013entropy, Appendix] for two different proofs of this result. See also [Yau, Donnelly, ohsawa2009remark, Ohsawa2, Treger, Yeung, ChenFu] for various related results and generalizations. The purpose of this work is to prove a more general version of this theorem, where the universal cover is replaced with any infinite Galois cover. Our results also make sense in genus .
It is known that the canonical form on the Poincaré unit disk is the same as the hyperbolic form (up to a constant multiple). So Kazhdan’s theorem can be restated, informally, as follows: the limit of the induced forms from finite Galois covers coincides with the induced form from the limiting space (the universal cover). This is the statement that we generalize.
1.2. Our results
The main purpose of this paper is to prove the following.
Theorem A. (A generalized Kazhdan theorem) Let be any infinite Galois cover of a compact connected Riemann surface of genus . Let be a sequence of finite Galois covers converging to . Then the sequence of -forms on induced from the canonical forms on converges uniformly to the -form induced from the canonical form on .
See Theorem 5.4 for a more precise statement.
Most of our work goes into the proof of a weaker result, stating that the associated measures attached to -forms converge strongly (Theorem 5.3). The uniform convergence of forms will then follow from a standard (but subtle) analytic argument.
We first directly define and study the notion of canonical measures on Riemann surfaces (Definition 3.1). These are precisely the measures attached to canonical -forms (Lemma 3.9), but they can be defined in more general situations (e.g. even if the surface is not orientable – see Remark 3.2 (iii)). The main reason we study these measures directly is that they are closely related to the operator theory on various natural Hilbert spaces attached to Riemann surfaces (see, e.g., Proposition 3.4). This will allow us to use powerful techniques from operator theory and the theory of von Neumann algebras.
A fundamental property of the hyperbolic measure on a compact Riemann surface is the well-known consequence of the Gauss–Bonnet formula, stating that the hyperbolic volume of a Riemann surface has a simple expression in terms of the Euler characteristic of . Our second main theorem states that all limiting measures (coming from any infinite Galois cover) satisfy a similar Gauss–Bonnet type property.
Theorem B. (A generalized Gauss–Bonnet) Let be an infinite Galois covering of a compact connected Riemann surface of genus , and let be the measure on induced from the canonical measure on . Then
Indeed, Theorem B is also a crucial ingredient in the proof of convergence of canonical measures (Theorem 5.3). Our proof shows that one may think of the equality as a ‘trace formula’. For the proof of Theorem B, we will need a slight variant of Lück’s approximation theorem ([Luck2, Theorem 0.1]) to deal with general infinite Galois covers (Theorem 4.6).
1.3. Related work and directions
Kazhdan’s theorem is usually stated in terms of the canonical metric, i.e. the metric whose associated -form is our canonical -form. This is a hermitian metric on the holomorphic tangent bundle on the Riemann surface. It is well-known that such a metric is directly recovered from its associated -form (see, e.g., [GriffHar, pp 28–29]). As we mostly work with the corresponding measure, we found it more convenient to state and prove our results in the language of the canonical forms. See also Remark 3.8 for some related notions and terminology in the literature.
Metric graphs may be considered as tropical (or non-Archimedean) analogues of Riemann surfaces. For metric graphs, analogues of Theorem A and Theorem B are proved in [ShokriehWu] by the second and third named authors. Following the discussion in [ShokriehWu, §7], we expect that the limiting measure inherited from the ‘Schottky cover’ of a Riemann surface to be closely related to other notions such as Poisson-Jensen and equilibrium measures. The limiting measure inherited from the maximal abelian cover should also have a nice intrinsic interpretation.
The starting point for our approach was the very slick proof of the classical Kazhdan’s theorem given by Curtis McMullen in [mcmullen2013entropy, Appendix] (based on the original argument in [Kaj, §3]). The main difficulty in our generalization is the basic fact that one knows explicitly the limiting measure on the universal cover (i.e. the hyperbolic measure) whereas, for an arbitrary infinite Galois cover, one does not have such explicit knowledge. Our use of and von Neumann algebraic techniques is to overcome this difficulty.
It is conceivable that our work can be generalized in various directions; there could be Kazhdan-type theorems for other types of limits of Riemann surfaces, most importantly Benjamini–Schramm limits in the sense of [ABBGNRS]. Moreover, one can formulate analogues of canonical measures and forms in higher dimensions. Our techniques can be adapted to prove analogues of Theorem A and Theorem B in higher dimensions as well. It is not clear, however, that one can obtain convergence of corresponding metrics in higher dimensions.
1.4. Structure of the paper
In §2 we review some basic facts and set the terminology and notations. In §3 we give the definitions for canonical measures and forms, and establish some of their basic properties. Our emphasis is to give operator theoretic interpretations and formulas. In §4 we first review some basics from the theory of von Neumann algebras and dimensions, as well as the Hodge–de Rham theory. The main purpose is to prove a variant of Lück’s approximation theorem (Theorem 4.6). In §5 we prove our main results (Theorem A and Theorem B).
We would like to thank John Hubbard, Bingbing Liang, Curtis McMullen, and Nicolas Templier for their interest in this project, and for helpful remarks and conversations. We would also like to thank Curtis McMullen and Nicolas Templier for their comments on an earlier draft. The first author was partially supported by Samsung Science & Technology Foundation grant no. SSTF-BA1702-01.
2. Notation and Background
2.1. forms on Riemann surfaces
Throughout, by a Riemann surface, we will mean a (possibly non-compact) connected Riemann surface. Our compact Riemann surfaces will always have genus .
Let be a Riemann surface. Let denote the Hilbert space completion of , the space of complex (global) -forms with compact support endowed with the hermitian inner product
where is the Hodge star operator. As usual, we use the notation .
Recall, on a Riemann surface, the Hodge star operator is defined by the local formula
In particular, it depends only on the complex structure and not on the choice of the Riemannian metric.
The space of holomorphic -forms (differentials of the first kind) and antiholomorphic -forms are orthogonal under the inner product in (1). For , one computes
where is the usual Hodge inner product on
The Hilbert space is separable. This is because the Riemann surface is second-countable and, consequently, the Borel -algebra is countably generated. Since the Riemann–Lebesgue measure is -finite on , it follows from [Cohn, Proposition 3.4.5] that admits a countable basis.
Let be an open subset. Then is itself a Riemann surface and we have a natural inclusion (extension by zero) of Hilbert spaces .
2.2. Convergence of measures and forms
We are mainly concerned with the measurable space consisting of a Riemann surface together with its Borel -algebra .
A sequence of measures on a measurable space is said to converge strongly to a measure if we have for every .
Given a nonnegative -form on a Riemann surface , the map
defines a measure on , which we denote by . Here, the set is considered together with the orientation inherited from the surface .
A sequence of nonnegative -forms on converges weakly to a -form if the sequence of associated measures converges strongly to the measure .
Fix a finite analytic atlas for a compact Riemann surface . We say a sequence of nonnegative -forms on converges uniformly to a -form if converges uniformly to on each coordinate chart . If is a local analytic coordinate in a domain and
we say converges uniformly to on if the sequence of real-valued nonnegative functions converges uniformly to on .
3. Canonical measures and forms
3.1. Canonical measures
Let be a Riemann surface, and be the subspace consisting of harmonic -forms:
where denotes the usual Hodge Laplacian (also known as the Laplace–de Rham operator) on -forms. It is self-adjoint with respect to the hermitian inner product in (1): if are supported on the interior of we have
Because the operator is elliptic, it follows from elliptic regularity (see, e.g., [evans1998partial, Chapter 6]) that harmonic -forms are indeed smooth. Moreover, form a Hilbert subspace of . This follow from, for example, the Hodge–de Rham decomposition ([luck, Theorem 1.57]).
Let be a Riemann surface. Let be an orthonormal basis for the Hilbert space . The canonical measure on is defined by
for any Borel subset .
If the surface is clear from the context, we will use instead of .
The index set is countable (Remark 2.2). Therefore is indeed a measure on , as it is a countable sum of integrals of smooth -forms.
If one uses the opposite orientation on , the forms become their negative and the measure remains unchanged. Our computations and arguments about can be generalized to the case where is not orientable.
The definition of is independent from the choice of the orthonormal basis .
If is a Galois covering map, the canonical measure on is the pullback of a measure on , which we shall henceforth denote by .
If is compact then .
The measure depends only on the complex structure on (and not on the particular choice of the Riemannian metric).
is invariant under conformal transformations.
(a) For a Borel subset , consider the nonnegative self-adjoint operator defined by . The trace of , with respect to the orthonormal basis , is:
The result now follows from the independence of trace from the choice of basis.
(b) It follows from the definition that is invariant under isometries.
(c) When is compact, we have .
(d) This follows from the fact that the Hodge star operator depends only on the complex structure (see (2)).
(e) This is the immediate consequence of part (d).
3.2. Properties of canonical measures
Some of the following properties of follow from the known properties of canonical forms (see §3.3), but we will give a self-contained treatment here. We begin by giving an alternate formula for the canonical measure of open sets in terms of orthogonal projections.
Let be a Riemann surface. Let be the orthogonal projection, and let be any open subset. Then
where is an orthonormal basis for the Hilbert subspace .
Let be an orthonormal basis for . Then
Let be a Riemann surface.
Assume is an open subset. For any open subset we have
Let be a sequence of open subsets with . Then, for any open subset ,
where and and denote the orthogonal projections.
(a) Because harmonicity is local, we have .
Let be the space consisting of -forms on which are also harmonic on (and therefore smooth on by elliptic regularity). Note that is a Hilbert subspace of . This is because is the orthogonal complement of the set (see (4)), and weak harmonic forms are the same as strong harmonic forms. Let be the orthogonal projection. Then
In other words, the orthogonal complement of in is the same as in . To see this, note that is the closest point in to , and is the closest point in to . But, for any , we have . Therefore must be the same as on , and must vanish otherwise.
(b) Let be the space of -forms on which are harmonic on . Again, is a Hilbert subspace of and its elements are smooth on . Since we have
where is the orthogonal projection. The last equality is because of the following fact: for any Hilbert space and a decreasing sequence of its Hilbert subspaces, the sequence of orthogonal projections onto subspaces converge pointwise to the orthogonal projection onto the intersection of those subspaces (this is an easy consequence of the Gram–Schmidt process and Parseval’s theorem). The result follows from (6) and (9).
Let be a Riemann surface.
is absolutely continuous with respect to the Riemann–Lebesgue measure.
is a Radon measure.
(a) We show that is absolutely continuous with respect to any Riemann-Lebesgue measure computed with any smooth Riemannian metric in the conformal class. It does not matter which Riemannian metric is used, because if two different Riemannian metrics result in two measures and then there is a smooth positive function so that , hence is absolutely continuous with respect to and vice versa.
Let be the Poincaré unit disk. An orthogonal basis of is
For any Borel subset we have, by Definition 3.1,
Therefore is absolutely continuous with respect to the Riemann–Lebesgue measure on the disk. For a general surface , around any point there is a conformally embedded open disk. Therefore the result follows from Proposition 3.5 (a).
(b) Since is a second countable, locally compact Hausdorff space, we only need to show that is finite on compact sets (see, e.g., [Folland, Theorem 7.8]). The Riemann-Lebesgue measure is locally finite and, by part (a), is absolutely continuous with respect to the Riemann-Lebesgue measure. Furthermore, from the computation in part (a), we see that the Radon–Nikodym derivative of with respect to the Riemann-Lebesgue measure is bounded from above by a locally bounded function. This implies that the is bounded on compact sets.
3.3. Canonical (Arakelov) forms
3.3.1. Compact Riemann surfaces
Let be a compact Riemann surface of genus . Fix an orthonormal (with respect to the Hodge inner product) basis for the vector space of (global) holomorphic -forms (differentials of the first kind) . Following [Arakelov, Faltings] (see also [LangAr, II,§2] or [deJong, §3]), the canonical (Arakelov) -form of is defined to be
It is easy to check (using Riemann–Roch theorem) that is indeed a volume form, and
On the Jacobian of there exists a canonical translation-invariant -form obtained by identifying with . One can easily check that is obtained by pulling back along the Abel–Jacobi map.
In Arakelov geometry, it is customary to study instead of . This normalization is not suitable in our context, as we will also deal with non-compact Riemann surfaces.
It is known that can be obtain as the Chern form of the canonical line bundle equipped with the Arakelov metric [Arakelov, §4]. For a description of the Arakelov metric in terms of Arakelov Green’s function see [LangAr, II,§2] or [deJong, §3].
The canonical metric on is, by definition, the hermitian metric on the holomorphic tangent bundle defined by (for the orthonormal basis as above). The canonical -form is the associated -form of , i.e. . It is well-known (see, e.g., [GriffHar, pp 28–29]) that uniquely determines .
Let the -form denote the Chern form of the pair . Let be the corresponding hermitian metric on the holomorphic tangent bundle . More precisely, and is the associated -form of . It can be checked that coincides with the pullback Fubini–Study metric under the canonical mapping .
In the literature, the term ‘Bergman metric’ sometimes refers to (see, e.g., [Neeman, mcmullen2013entropy]), and sometimes refers to (see, e.g., [mumford, rhodes1993sequences]).
3.3.2. General Riemann surfaces
Let be a (possibly non-compact) Riemann surface. Fix an orthonormal basis (with respect to the Hodge inner product) for the Hilbert space of holomorphic -forms. The canonical (Arakelov) -form of is defined to be the nonnegative -form
Recall (see §2.2) the map defines a measure on .
Let be an orthonormal basis (with respect to the Hodge inner product) for the vector space of holomorphic -forms. An orthonormal basis (with respect to the product (1)) for is . For a Borel set , by Definition 3.1, we have
Note that and for any .
There is also a nice local description for which we now describe (see [mcmullen2013entropy, Appendix]). Consider any holomorphic local chart , where denotes the -disk around . The canonical -form can be expressed on in terms of the coordinate as
for some real-valued nonnegative function . If is an orthonormal basis of the space of holomorphic -forms on , and in terms of the coordinate , then .
Let be an orthonormal basis of the space of holomorphic -forms on , and let in terms of the coordinate . We have
The second equality is by Cauchy–Schwarz.
4. A variant of Lück’s approximation
4.1. Hilbert -modules and -dimensions
Here we quickly review the von Neumann algebras and dimensions that appear in our context. See [luck] for proofs and a more thorough treatment.
For any complex Hilbert space , let denote the algebra of all bounded linear operators on , and let . By a Hilbert -module we mean a Hilbert space together with a (left) unitary action of the discrete group . A free Hilbert -module is a Hilbert -module which is unitarily isomorphic to , where is a Hilbert space with the trivial -action and the action of on is by left translations. Let be an orthonormal basis for . Then we have an orthogonal decomposition
where is a copy of .
Recall, for each , we have the left and right translation operators defined by , for . We are interested in the von Neumann algebra on , where is the von Neumann algebra generated by . Alternatively is the algebra of -equivariant (i.e. those commuting with all ’s) bounded operators on .
Let be an orthonormal basis for . For every , define
for a fix . Here denotes the indicator function of . This is independent of the choice of , so one usually picks , the group identity. It can be checked that this is a “trace function” in the sense of von Neumann algebras.
A projective Hilbert -module is a Hilbert -module which is unitarily isomorphic to a closed submodule of a free Hilbert -module, i.e. a closed -invariant subspace in some . Note that the embedding of into is not part of the structure; only its existence is required. Fix such an embedding. Let denote the orthogonal projection from onto . Then because it commutes with all . The -dimension of is defined as
An elementary fact is that does not depend on the choice of the embedding of into a free Hilbert -module; it is a well-defined invariant of .
-dimensions satisfy the following expected properties:
. Equality holds if and only if .
If is a short weakly exact sequence of projective Hilbert -modules, then .
and are weakly isomorphic, then .
A sequence of of projective Hilbert -modules is called weakly exact at if . A map of projective Hilbert -modules is a weak isomorphism if it is injective and has dense image.
4.2. Hodge–de Rham theorem
Let is a Galois covering of a finite CW complex , with as the group of deck transformations.
Let denote its cellular -cochain complex:
where is the usual cellular chain complex, considered as a -module. Fixing a cellular basis for , one obtains and explicit isomorphism
for some integer . Therefore has the structure of a projective Hilbert -module. Let denote the induced -coboundary map. The (reduced) -th -cohomology of the pair is defined by
Note that, since we divide by the closure of the image, the resulting inherits the structure of a Hilbert space. It is, moreover, a projective Hilbert -module because . Therefore it makes sense to define the -th -Betti number of the pair by
The following version of Hodge-de Rham theorem is proved in [Dodziuk] (see also [luck, Theorem 1.59]).
Theorem 4.3 ( Hodge–de Rham theorem).
Let be a Galois covering of a compact Riemannian manifold , with as the group of deck transformations. Assume further has no boundary. Let be an equivariant smooth triangulation of . Then there is a canonical isomorphism (as projective Hilbert -modules) between the space of harmonic smooth -forms and the -th -cohomology of the pair :
Furthermore, is finite.
We only use this theorem when and are Riemann surfaces and . In this case, the space of harmonic smooth -forms is denoted by in (3). General spaces are defined analogously in higher dimensions.
It is well known that if is a Riemann surface of genus we have . This follows from [luck, Theorem 1.35] (see also [luck, Example 1.36]). Alternatively, one can deduce this from our Thoerem 4.6.
4.3. Approximation theorem
Here we will prove a variant of Lück’s approximation theorem in [Luck2]. Let be a discrete group as before. Let be a -module homomorphism. After tensoring with and completion, we obtain an induced map
For any finite index normal subgroup , we also have a map
induced by the quotient maps and tensoring with . Concretely, if denotes the standard matrix of with respect to the standard bases for and , then also represents the standard matrix of and of . In this situation, we have the following beautiful theorem of Lück (see [Luck2, Theorem 2.3]):
Let be a descending sequence of finite index normal subgroups of such that . Then,
The following result is an appropriate modification of [Luck2, Theorem 0.1], needed for our application.
Let be a finite connected CW complex. Let be an infinite Galois covering. Let be an ascending sequence of finite Galois coverings converging to , in the sense that the equality
holds in . Let
denote the deck transformation group the covering ,
denote the degree of the covering ,
denote the (ordinary) cohomology of .
Our proof is a modification of Lück’s proof of [Luck2, Theorem 0.1]. There, the proof is given for the special case where is the universal cover of .
Let be the (ordinary) coboundary map on . Then is the -coboundary map on , and will be the (ordinary) coboundary map on .
Let . Then is a descending sequence of subgroups of . Moreover implies that . Clearly, and, by Theorem 4.5, we obtain:
Let be the number of -cells in . Then
is isomorphic to . By Remark 4.2 (ii) and (iv) we have:
is a free -module of rank